For questions on dice, a small throwable object with multiple resting positions, used for generating random numbers. This makes dice suitable as gambling devices for games like craps, or for use in non-gambling tabletop games.

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Throwing dice and finding limits

We throw an fair dice $n$ times. Let $S_n$ be the number of throws with even number of dots on the dice. 1.) Calculate the limit $$\lim_{n\rightarrow\infty}P(2S_n \leq n)$$ 2.) Express the value of ...
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Estimating a random variable from repeated trials

I have an $n$ sided die and suspect that it is biased. I'm interested in the probability of rolling a $1$, so I roll the die $m$ times and count up the number of times I roll $1$, then divide the ...
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27 views

Figuring out probability of dice with least amount of questions

Given $n$ dice, each with $k$ faces numbered $1,\dots,k$, you're allowed to ask me what the probability of some event happening is (a subset of all the possibilities and I give a number). What ...
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32 views

Probability: Application Of “Expected Value”

$\newcommand{\P}{\mathbb{P}}$$\newcommand{\E}{\mathbb{E}}$So, I was learning expected value today and I'm trying to understand the significance of calculating this term "Expected value". In this ...
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30 views

Confidence interval when rolling a dice

If you roll a die 20,000 times and from these rolls you get either a 1,2, or 3 11,000 times, how can you calculate the confidence interval at 95% for the probability to get a 1,2, or 3? The ...
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“In tossing four fair dice, what is the probability of at least one three?” using complement

There is a very similar question here, and in both that question and this question I'm asking right now, the accepted answer involves using a complement. The same thing occurs in this textbook ...
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Probability sum of 5 before sum of 7

Pair of fair die are rolled (independently I hope) infinitely. Find probability sum of 5 appears before sum of 7. 2 approaches: $$P(\text{sum of 5 appears before sum of 7})$$ $$= P(\text{roll 1 ...
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39 views

What is the average and variation of $20$ dices?

If I roll a dice the average is $E(X) = (1+2+3+4+5+6)/6 = 7/2$ and $$E(X^2) = (1+4+9+16+25+36)/6 = 91/6$$ $$VAR(x) = E(X^2) - (E(X))^2 = 91/6 - 49/4 = 35/12$$ Now the question is: How I can find ...
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80 views

Find $\mathbb P (X_1 + \cdots + X_n = 6n-3)$

A fair die is tossed n times (for large n). Assume tosses are independent. What is the probability that the sum of the face showing is $6n-3$? Is there a way to do this without random variables ...
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1answer
23 views

Probability of Getting a Yahtzee of Fives Given Two Fives

(The following problem is from MAML, Meet 3, Round 1, December 2012, Problem 3.) In the game of Yahtzee one has a chance to get Yahtzee (5 of the same kind, such as 5 sixes) in the throw of 5 ...
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Probability of getting a “full house” by rolling dice [duplicate]

In poker, full house means getting three cards with the same rank, and another two cards with the same rank (not the same as other three cards). I can understand how to use combination to solve this ...
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27 views

Calculating Probability Of Value With Dice [duplicate]

I am trying to write a program to calculate the probability of a number of dice thrown equaling a specif value. I have done some working out in excel, to try and find a patter, but I am at a loss. At ...
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2answers
32 views

Throwing x dice, chance of getting sum of at least y [duplicate]

I found some answers to this question, but that was with only two dice, with 6 sides. What if I have 30 dice, with 10 sides each (from 1 to 10). I can't make a matrix and count as was suggested on the ...
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1answer
39 views

Quick probability question on rolling 2 Die

If I flip a coin and it lands on heads I roll a fair dice n times. If the coin lands on tails I roll a biased dice n times. Let $X_i$ denote the score of the $i$th roll of the fair die and $Y_i$ the ...
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12 six-sided dice are thrown, what is the probability of getting each number at least once? [duplicate]

I got a bit confused and couldn't find the correct answer to check mine. This was the closest I could get: $\frac{6^6 \frac{12!}{6!6!}}{6^{12}}=\frac{924}{6^6}=0,0198$ where you have $12 \choose6$=$...
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Probability to obtain more than X with 3 dice.

This is a dice problem 1) I want to calculate the probability to have more than X throwing 3 dice of 6 faces. X = addition of the result of the 3 dice. 2) This is the first step but if you can also ...
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1answer
48 views

Probability Mass Function of infinitely re-rolled dice

I play a game called Shadowrun. It is a role-playing game that uses a dice pool mechanic. A player has a dice pool of $x$ six-sided, unbiased dice. Every 5 or 6 counts as a success. The more successes,...
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52 views

The probability of rolling 4 dice and getting a 6.

The probability of rolling 2 dice and getting a 6 on either one of the die or both is : 11/36 or about 0.305. Also I calculate the probability of rolling 4 dice and getting a 6 on either one, two, ...
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2answers
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Dice role: What is the probability to observe 2 times 1 and 2 times 5 with the outcome of a fifth die role being unknown?

I tried to solve the following exercise: Given a dice with $P(X=2) = P(X=4) = P(X=5) = \frac{2}{15}$ and $P(X=1) = P(X=6) = P(X=3) = \frac{2}{10}$. What is the probability to observe 2 times 1 and 2 ...
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1answer
18 views

Probability of rolling a five and a prime pair of six sided dice

This is a silly question, but I can't quite put my finger on where my reasoning is wrong. Given a pair of 6 sided dice, what is the probability of rolling a 5 and a prime? My answer: 6/36 Correct ...
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67 views

Generalization of classic 3 roll die game to $n$ rolls

I am trying to generalize the following well-known 3 roll die problem: "We roll a single die no more than 3 times. We can stop immediately after the first roll, immediately after the second roll, or ...
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Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
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1answer
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Any way to calculate chances of getting “n” hits when rolling “x” die (hit is when I roll more than “y”)?

First, let me preface - I saw similar questions already, but to be honest, I didn't understand the answers, or couldn't understand how to convert the answer for given question to my problem. My ...
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Probability of the n-th dice matching in 2 groups of sorted random numbers

Say we have 2 groups of 6 6-sided dice. Each group of dice is rolled and then sorted so we have 2 groups of sorted numbers. What is the probability of each die in the group matching the corresponding ...
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Two regular dice are rolled. Given that one of them is a $6$, what is the probability that the other is also a $6$?

I'm having trouble with the following question for and I have an exam in two days: Two regular dice are rolled. Given that one of them is a $6$, what is the probability that the other is also a $6$...
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2answers
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Proof: Probability of a pair when rolling 7 dice is 1.

I know that the probability should be over or exactly 1 since out 6 possible values the 7th dice will always be a duplicate. My calculations are wrong though: $\frac{{7\choose2} 5! 6*1}{6^7}$ Why?...
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46 views

Probability of disjoint dice results

You have $n = n_A + n_B$ $k$-sided dice. The $n_A$ dice are thrown and a set of the resulting values, call it $S_A$, is built; likewise for the $n_B$ dice, calling the resulting set $S_B$. What is ...
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1answer
41 views

What is the probability that the sum of two dice rolls is a multiple of $3$?

What is the probability that the sum of $2$ dice rolls is a multiple of $3$? What about for $3$ dice rolls? For $n$ dice rolls? So I have the first part of this solution worked out by writing out all ...
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1answer
59 views

Optimal bet according to the probability of win

Suppose the following game: You start with $5000; You will roll a dice 100 times; You have to choose a percentage of your accumulated money to bet all the 100 times. You will choose it only once and ...
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Dice throwing probability : at least one success

What would be the function for calculating the probability of at least one success of $n$ $10$-sided dice thrown, if success is $9$ or $10$, but $1$ is counted as negative success (or failure). In ...
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390 views

Die that never rolls the same number consecutively

Suppose we have a "magic" die $[1-6]$ that never rolls the same number consecutively. That means you will never find the same number repeated in a row. Now let's suppose that we roll this die $1000$ ...
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Special dice generating non-decreasing sequence

Suppose that, when rolled for the first time, a special 6-sided dice shows $1,\ldots, 6$ with probability $\frac{1}{6}$ each, and then, upon rerolling, shows with equal probability a number greater or ...
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Calculating probability that 5 dice rolls will produce an average of 3.5

We are carrying out a dice game which involves 5 people rolling dice and passing buttons equal to the number rolled . We have to work out the probability that we will get an average of 3 .5 items ...
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Dice probability, Heaven or Hell [closed]

You're at the gate of Heaven and St. Peter asks you to play Dice with him. If you roll a 5 or 6, he'll let you into Heaven. However, if you roll a 2,3 or 4, you'll be sent to hell. If you roll a 1, he ...
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Find dependent event when two dice are thrown simultaneously.

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of ...
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Can someone help me balance this game (probability question) [closed]

A team of 9 vs a team of 1. Each round each of "the 9" roll a die to "attack" and "the 1" rolls 9 dice to "defend", the nine dice are preassigned to attackers before the roll, "the 1" cannot choose ...
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1answer
33 views

Given three six-sided dice, what is the probability that the value of the third will be greater than the sum of those of the first two?

A die is rolled three times, or three dice are rolled. What is the probability that the third die values greater than the sum of the first two? (assuming six-sided dice, but I would be interested in ...
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Creating $Y \sim U[1, \dots, 6] + U[1, \dots, 6]$ as a function of $X_1, X_2, X_3 \sim U[1, \dots, 6]$

The Three Indistinguishable Dice Puzzle from standupmaths The problem therein can be summarized as follows: You have three dice rolls. Each die is indistinguishable from the others. However, using ...
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You cast a pair of dice. If you get a 6 and an 8 before 7 comes up twice, you win. What is the probability of winning?

You cast a pair of dice. If you get a 6 and an 8 before 7 comes up twice, you win. What is the probability of winning? What I tried was \begin{align*} P(X=6) & = \frac{5}{36}\\ P(X=8) & = ...
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4answers
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Rolling a die until two rolls sum to seven

Here's the question: You have a standard six-sided die and you roll it repeatedly, writing down the numbers that come up, and you win when two of your rolled numbers add up to $7$. (You will ...
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What is the probability of these questions?

What is the probability of throwing $2$ dice, and the first one is a $6$? (Given that one of them is a 6.) My solution says that it's $\frac{6}{11}$, but I have no clue why. The other one is: What ...
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1answer
678 views

How to use N indistinguishable dice to simulate 1 roll of N-1 dice? [closed]

If I use 3 indistinguishable dice to simulate 1 roll of 1 die. I can use this formula "(SUM % 6) + 1". There SUM is the total amount of the three indistinguishable dice added together, 6 is the max ...
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Probability of three dice falling in the same quadrant of a box

This is surely really basic for most people here but it's tripping me up. You get a box and draw lines to split it up into 4 parts. I got asked what the probability was that when rolling three dice,...
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Roll two fair dice, what is the expected value of rolls such that the sum of the two faces is the multiple of 6.

I was thinking that there were 2 cases: (1) $\\$ $X_{1}+X_{2}=6$, and the probability would be $\dfrac{5}{36}$. (2) $\\$ $X_{1}=X_{2}=6$, and the probability would be $\dfrac{1}{36}$. I am now ...
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1answer
51 views

Probability of being able to make a given number with two out of xd6

I'm an amateur games designer, and am working on a mechanic which involves rolling on a table with numbers running from 2-12 - the full range of possibilities from adding together two six-sided dice. ...
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2answers
42 views

Rolling dice against each other?

After doing some research, I found that the probability of rolling an n-sided die against another, equivalent die and landing a higher number on the first rolled dice to be ((n−1)/2)/n, where n is the ...
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Probability of Rolling a $1$ on an $n$-Die in $r$ Attempts

I roll an $n$ sided die, numbered $1 \to n$. If I roll a $1$, I walk away; otherwise, I roll the die again. This process could repeat indefinitely. What is the probability, $P(n,r)$, that I will roll ...
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7th Sea 2e: Grouping dice results into sets of 10 vs doing the same with Tarot cards.

Note: While an interesting problem on its own, it also struck me as an excellent example of how marvelous the human brain is, as it is able to almost effortlessly negotiate a task that (at least to ...
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4answers
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Probability of obtaining a heads on the coin before a 1 or 2 on the die?

I came across this question recently and can't seem to find the correct approach. Any help would be appreciated! An experiment consists of first tossing an unbiased coin and then rolling a fair ...
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1answer
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Probability of rolling dice twice

Did I calculate the correct probability for these simple scenarios: 1: What is the probability of rolling 3 and 4 with two dice in two rolls? If you roll either 3 or 4 in the first roll, you put ...